Tubes do have more distortion than solid state, but well engineered tube amps driving appropriate loads don't have high enough distortion to become audible. At least that was my experience owning the Wheatfield HA-2, a high quality SET OTL tube headphone amp. But this tube amp did sound warmer than a solid state amp with most headphones. Most likely due to its high output impedance combined with most headphones having peak impedance in the bass, which acts as a tone control. You can mimic that with EQ.
Even the output impedance effect isn't unique to tube amps. My current solid state headphone amp has 2 headphone outputs each with different output impedances. One is zero ohms, the other 120 ohms. With dynamic headphones like the Sennheiser HD-580 you can quick switch back & forth and hear the difference in tonality (warmer sound from higher output impedance). If you quick switch planar headphones, the 120 ohm output doesn't change the tonality, it just gets quieter. That's because planars have flat impedance vs. frequency.
Incidentally, I've read that decades ago the 120 ohm output impedance for headphones was considered 'standard' and some headphones were designed for this. They sound too bright on modern gear, relying on the amp's high output impedance to tame the treble.
All else equal, if you can get the same euphonic coloration by (A) purchasing gear that does it inherently, or (B) purchasing transparent gear and applying the euphonics through EQ, it's obvious that (B) is the better choice. The real question then becomes: can one truly reproduce that "tubuliciuos" sound by applying EQ or other DSP with transparent solid state equipment?